Average Error: 52.6 → 0.2
Time: 25.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0776022250282242:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.955290469990551:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0776022250282242:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.955290469990551:\\
\;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\

\end{array}
double f(double x) {
        double r6415484 = x;
        double r6415485 = r6415484 * r6415484;
        double r6415486 = 1.0;
        double r6415487 = r6415485 + r6415486;
        double r6415488 = sqrt(r6415487);
        double r6415489 = r6415484 + r6415488;
        double r6415490 = log(r6415489);
        return r6415490;
}
↓
double f(double x) {
        double r6415491 = x;
        double r6415492 = -1.0776022250282242;
        bool r6415493 = r6415491 <= r6415492;
        double r6415494 = -0.0625;
        double r6415495 = r6415491 * r6415491;
        double r6415496 = r6415495 * r6415491;
        double r6415497 = r6415495 * r6415496;
        double r6415498 = r6415494 / r6415497;
        double r6415499 = 0.125;
        double r6415500 = r6415499 / r6415491;
        double r6415501 = r6415500 / r6415495;
        double r6415502 = 0.5;
        double r6415503 = r6415502 / r6415491;
        double r6415504 = r6415501 - r6415503;
        double r6415505 = r6415498 + r6415504;
        double r6415506 = log(r6415505);
        double r6415507 = 0.955290469990551;
        bool r6415508 = r6415491 <= r6415507;
        double r6415509 = 0.075;
        double r6415510 = r6415509 * r6415497;
        double r6415511 = -0.16666666666666666;
        double r6415512 = r6415496 * r6415511;
        double r6415513 = r6415510 + r6415512;
        double r6415514 = r6415513 + r6415491;
        double r6415515 = r6415491 + r6415491;
        double r6415516 = r6415501 - r6415515;
        double r6415517 = r6415503 - r6415516;
        double r6415518 = log(r6415517);
        double r6415519 = r6415508 ? r6415514 : r6415518;
        double r6415520 = r6415493 ? r6415506 : r6415519;
        return r6415520;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.6
Target45.0
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0776022250282242

    1. Initial program 62.0

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around -inf 0.1

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

    3. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right) + \frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\right)}\]

    if -1.0776022250282242 < x < 0.955290469990551

    1. Initial program 58.7

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

    3. Simplified0.2

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left(\left(x \cdot x\right) \cdot x\right) + \frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)\right) + x}\]

    if 0.955290469990551 < x

    1. Initial program 30.4

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]

    2. Taylor expanded around inf 0.3

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{8} \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.3

      \[\leadsto \log \color{blue}{\left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0776022250282242:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.955290469990551:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))